Collaborative Research: Development of High-Resolution Finite-Volume Methods for Systems of Nonlinear Time-Dependent PDEs

合作研究：非线性时变偏微分方程组高分辨率有限体积方法的开发

• 批准号: 1115718
• 负责人: Alexander Kurganov
• 金额: $ 11.86万
• 依托单位: Tulane University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-10-01 至 2015-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1115718&HistoricalAwards=false
• 关键词: Collaborative; Research; Development; Resolution; Finite

The project is aimed at developing highly accurate, efficient and robust numerical methods for systems of nonlinear time-dependent PDEs, with particular reference to multidimensional hyperbolic systems of conservation/balance laws and related problems. The principal part of the proposed research will be focused on the development of new finite-volume methods that will provide an improved resolution of linear contact waves and incorporate new techniques for solving problems involving complicated nonlinear wave phenomena and blowing up/spiky solutions. The proposed methods will be applied to a variety of nonlinear problems, among which are systems of gas dynamics, nonlinear elasticity and acoustics systems, modern traffic flow models, several chemotaxis and bioconvection models, and others. These problems will be studied in the most challenging cases of high space dimensions, complex geometries and moving interfaces. For each problem, a high-resolution finite-volume scheme will be systematically derived in a way that the main properties satisfied by the underlying system of PDEs will be also satisfied on the discrete level. One of the key features of the new schemes will be their nonlinear stability, which will be ensured by ability of the scheme to preserve positivity of such physical quantities as density. To achieve this goal, several high-order positivity preserving techniques will be explored.Besides providing the examples that corroborate the analytical approach, the foregoing applications are of a substantial independent value for a broad class of problems arising in today's science including geophysics, meteorology, astrophysics, semiconductors, traffic flows, image processing, financial and biological modeling and many other areas. Development of modern high-resolution finite-volume methods as well as of supplementary techniques is essential for solving many practically important problems, some of which are currently out of reach because the existing numerical methods are either inefficient/inaccurate or not applicable at all.

该项目旨在为非线性时间依赖性PDE的系统开发高度准确，高效且健壮的数值方法，特别是参考了多维纯金保护/平衡定律和相关问题的多维双皮系统。拟议的研究的主要部分将集中于新的有限体积方法的开发，该方法将改善线性接触波的分辨率，并结合了解决涉及复杂非线性波现象和爆炸/尖刺解决方案的新技术。所提出的方法将应用于各种非线性问题，其中包括气体动力学系统，非线性弹性和声学系统，现代交通流模型，几种趋化性和生物传染模型等。这些问题将在最具挑战性的高空间维度，复杂几何形状和移动界面的情况下进行研究。对于每个问题，高分辨率有限体积方案将系统地得出，以使PDE基础系统满足的主要属性也将在离散级别上满足。新方案的关键特征之一将是它们的非线性稳定性，该计划将通过该方案保持阳性诸如密度等物理量的能力来确保。为了实现这一目标，将探索几种高阶阳性保存技术。Besides提供了证实分析方法的示例，上述应用对于当今科学中引起的广泛的问题具有实质性的独立价值，包括地球物理学，包括气象学，天体物理学，分类流量，流量流动，图像处理，金融和其他模型和其他模型和其他模型和其他模型。开发现代高分辨率有限体积方法以及补充技术对于解决许多实际重要的问题至关重要，其中一些问题目前已经遥不可及，因为现有的数值方法效率低下/不准确或根本不适用。